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The precise analysis of the performance of a disjoint-set forest is somewhat intricate. However, there is a much simpler analysis that proves that the amortized time for any m Find or Union operations on a disjoint-set forest containing n objects is O(m log * n), where log * denotes the iterated logarithm. [12] [13] [14] [15]
The pseudocode below determines the lowest common ancestor of each pair in P, given the root r of a tree in which the children of node n are in the set n.children. For this offline algorithm, the set P must be specified in advance. It uses the MakeSet, Find, and Union functions of a disjoint-set data structure.
An efficient implementation using a disjoint-set data structure can perform each union and find operation on two sets in nearly constant amortized time (specifically, (()) time; () < for any plausible value of ), so the running time of this algorithm is essentially proportional to the number of walls available to the maze.
By definition of wandering sets and since preserves , would thus contain a countably infinite union of pairwise disjoint sets that have the same -measure as . Since it was assumed μ ( X ) < ∞ {\displaystyle \mu (X)<\infty } , it follows that A {\displaystyle A} is a null set, and so all wandering sets must be null sets.
In mathematics, the disjoint union (or discriminated union) of the sets A and B is the set formed from the elements of A and B labelled (indexed) with the name of the set from which they come. So, an element belonging to both A and B appears twice in the disjoint union, with two different labels.
Many examples of presheaves come from different classes of functions: to any , one can assign the set () of continuous real-valued functions on . The restriction maps are then just given by restricting a continuous function on U {\displaystyle U} to a smaller open subset V ⊆ U {\displaystyle V\subseteq U} , which again is a continuous function.
The particular point topology on any infinite set is locally compact in senses (1) and (3) but not in senses (2) or (4), because the closure of any neighborhood is the entire space, which is non-compact. The disjoint union of the above two examples is locally compact in sense (1) but not in senses (2), (3) or (4).
The concept of disjoint union secretly underlies the above examples: the direct sum of abelian groups is the group generated by the "almost" disjoint union (disjoint union of all nonzero elements, together with a common zero), similarly for vector spaces: the space spanned by the "almost" disjoint union; the free product for groups is generated ...